Can quantum fluctuations affect the double pendulum?

On real physical scales quantum effects are nowhere near affecting classically chaotic systems. The point of chaos is that there is going to be some fluctuation, be it an imperfection of your model, thermal oscillations, a small whirl of air... which very early causes your trajectory to drift away from the one you predict. And there is no difference in whether the fluctuation is quantum or classical.

However, you can find microscopical systems for which classical equations of motion would exhibit chaos but they are actually quantized. The field studying the properties of such systems is called Quantum chaos. The quantum equations are linear which means that they cannot exhibit chaos and the chaos we see is only a property of the averaged "classical" picture of the dynamics. It actually turns out that the delocalization or "smearing out" of tiny deviations in initial condition typical for chaos gets often supressed in a classically-chaotic dynamical system under quantization. (But on time-scales of order $1/hbar^2$ where $hbar$ is the Planck constant.)

So back to your question: you should really worry more about the classical perturbations because they are much larger. If you insist on adding the "quantum" into chaos, things get much more complicated and unintuitive effects emerge which cannot be really explained by talking about quantum fluctuations.